Problem: Find $\lim_{x\to\infty}\dfrac{x^2-4}{\cos(x)}$. Choose 1 answer: Choose 1 answer: (Choice A) A $4$ (Choice B) B $-2$ (Choice C) C $0$ (Choice D) D The limit doesn't exist
Answer: When dealing with limits that include $\cos(x)$, it's important to remember that $\lim_{x\to\infty}\cos(x)$ doesn't exist, as $\cos(x)$ keeps oscillating between $-1$ and $1$ forever. ${2}$ ${4}$ ${6}$ ${8}$ ${\llap{-}4}$ ${\llap{-}6}$ ${\llap{-}8}$ ${2}$ $y$ $x$ $y=\cos(x)$ This doesn't necessarily mean that our limit doesn't exist. Think what happens to $\dfrac{x^2-4}{\cos(x)}$ as $x$ increases towards positive infinity. While $x^2-4$ keeps growing boundlessly, $\cos(x)$ oscillates from $-1$, to $0$, to $1$, to $0$, to $-1$ again. The result is a graph that goes up and down forever, with vertical asymptotes every now and then. ${5}$ ${10}$ ${15}$ ${\llap{-}5}$ ${\llap{-}10}$ ${\llap{-}15}$ ${50}$ ${100}$ ${150}$ ${\llap{-}50}$ ${\llap{-}100}$ ${\llap{-}150}$ $y$ $x$ $y=\dfrac{x^2-4}{\cos(x)}$ This limit doesn't approach any specific value as $x$ increases towards infinity. In conclusion, $\lim_{x\to\infty}\dfrac{x^2-4}{\cos(x)}$ doesn't exist.